Research of Algebras relating to Cartan Problem by using methods of Representation Theory as an application of Algebraic Geometry

• Project/Area Number: 19540019
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: University of Yamanashi
• Principal Investigator: SATO Masahisa University of Yamanashi, 大学院・医学工学総合研究部, 教授 (30143952)
• Co-Investigator(Kenkyū-buntansha): MORI Izuru 静岡大学, 理学部, 准教授 (50436903) ; OHNUKI Yousuke 鈴鹿工業高等専門学校, 数学教室, 講師 (30435432) ; NAKAMOTO Kazunori 山梨大学, 総合分析センター, 教授 (30342570) ; 若松 隆義 埼玉大学, 教育学部, 教授 (00192435)
• Project Period (FY): 2007 – 2009
• Project Status: Completed (Fiscal Year 2009)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2009: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2008: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2007: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
• Keywords: 環論 / 多元環 / 表現論 / 代数幾何学 / カルタン / 代数学

Research Abstract

代数閉体上の有限表現型遺伝多元環上の直既約加群はディンキン図形の正のルートと対応している。負のルートについては、加群上では、何を意味しているのか全く不明であった。本研究により、導来圏の対象と対応させていくことで、加群の複体のホモトー同値類、即ち導来圏の対象と代数的構造も含めて対応していることがわかった。また、カルタン行列は、導来圏に対する不変量であり、カルタン行列から作られるコクセター行列が、導来圏の対象において、アウスランダー・ライテン変換を与えるものであることがわかった。

Research Products

• [Journal Article] カルタン行列とその周辺の話題について (2009)
  • Author(s): 佐藤眞久
  • Journal Title: 第54回代数学シンポジューム報告集 Pages: 1-29
• [Journal Article] Quantum ruled surfaces defined by quivers (2009)
  • Author(s): Izuru Mori
  • Journal Title: 第54回代数学シンポジューム報告集 Pages: 65-75
  • NAID: 120005763714
• [Journal Article] Asymmetry of Ext-groups (2009)
  • Author(s): Izuru Mori
  • Journal Title: J. Algebra 322 Pages: 2235-2250
  • NAID: 120001788215
  • Peer Reviewed
• [Journal Article] On the classification of decomposable quantum rule surfaces (2009)
  • Author(s): Izuru Mori
  • Journal Title: Ring Theory 2007 Pages: 126-140
  • Peer Reviewed
• [Journal Article] The work of Tachikawa on Ring Theory in the world (2009)
  • Author(s): Masahisa Sato
  • Journal Title: Ring Theory 2007 Pages: 8-14
  • Peer Reviewed
• [Journal Article] Asymmetry of Ext-groups (2009)
  • Author(s): Izuru Mori
  • Journal Title: J.Algebra 322 Pages: 2235-2250
  • NAID: 120001788215
  • Peer Reviewed
• [Journal Article] The moduli spaces of non-thick irreducible representations for the free group of rank 2 (2009)
  • Author(s): Kazunori Nakamoto
  • Journal Title: 第41回環論および表現論報告集 41 Pages: 55-62
• [Journal Article] On the Classification of Decomposable Quantum Ruled Surfaces (2009)
  • Author(s): Izuru Mori
  • Journal Title: Ring Theory 2007 Pages: 126-140
  • Peer Reviewed
• [Journal Article] Intersection theory over quantum ruled surfaces (2008)
  • Author(s): Izuru Mori
  • Journal Title: J. Pure Appl. Algebra 211 Pages: 25-41
• [Journal Article] Co-point Modules over Frobenius Koszul Algebras (2008)
  • Author(s): Izuru Mori
  • Journal Title: Comm. Algebra 36 Pages: 4659-4677
  • Peer Reviewed
• [Journal Article] An Introduction to Noncommutative Algebraic Geometry (2008)
  • Author(s): Izuru Mori
  • Journal Title: Proc. Japan Ring Theory 36 Pages: 53-59
• [Journal Article] The moduli spaces of non-thick irreducible representtations for the free group of rank 2 (2008)
  • Author(s): Kazunori Nakamoto
  • Journal Title: Proc. Japan Ring Theory 36 Pages: 55-62
• [Journal Article] The links of 3-dimensional singularities defined by Brieskorn polynomialss (2008)
  • Author(s): Kazunori Nakamoto
  • Journal Title: Math. Nachr. 12 Pages: 1777-1790
  • Peer Reviewed
• [Journal Article] The links of 3-dimensional singularities definedby Brieskorn polynomialss (2008)
  • Author(s): Kazunori Nakamoto
  • Journal Title: Math. Nachr. 12 Pages: 1777-1790
  • Peer Reviewed
• [Journal Article] An Introduction to Noncommutative Algebraic Geometry (2008)
  • Author(s): Izuru Mori
  • Journal Title: Proceedings of the 40th Symposium on Ring Theory and Representation Theory 36 Pages: 53-59
• [Journal Article] Symmetry in the vanishing of Ext over stably symmetric algebras (2007)
  • Author(s): Izuru Mori
  • Journal Title: J. Algebra 310 Pages: 708-729
  • Peer Reviewed
• [Journal Article] Symmetry in the vanishing of Ext-group (2007)
  • Author(s): Izuru Mori
  • Journal Title: Proc. Japan Ring Theory 35 Pages: 79-86
• [Journal Article] symmetry in the vanishing of Ext over stably symmetricalgebras (2007)
  • Author(s): Mori, Izuru
  • Journal Title: J. Algebra 310 310 Pages: 708-729
  • Peer Reviewed
• [Journal Article] Intersection theory over quantum ruled surfaces (2007)
  • Author(s): Mori, Izuru
  • Journal Title: J. PureAppl. Algebra 211 Pages: 25-41
  • Peer Reviewed
• [Presentation] Classification problems in non commutative algebraic geometry and representation theory (2010)
  • Author(s): 毛利出
  • Organizer: 日本数学会2 010年年会代数学分科会特別講演
  • Place of Presentation: 慶応義塾大学、神奈川
  • Year and Date: 2010-03-25
• [Presentation] Classification Problems in non commutative algebraic geometry and representation theory (2010)
  • Author(s): 毛利出
  • Organizer: 日本数学会2010年年会代数分科会特別講演
  • Place of Presentation: 慶応義塾大学、神奈川
  • Year and Date: 2010-03-25
• [Presentation] カルタン行列とその周辺の話題について (2009)
  • Author(s): 佐藤眞久
  • Organizer: 第54回代数学シンポジューム
  • Place of Presentation: 明治大学、東京
  • Year and Date: 2009-08-03
• [Presentation] B-construction and C-construction (2008)
  • Author(s): Izuru Mori
  • Organizer: Special Session on Quantum Algebras and Related Topics-Joint Meeting of the AMS and SMS
  • Place of Presentation: Fudan University, Shanghai, China
  • Year and Date: 2008-12-18
• [Presentation] Homological Properties of Connected Graded Algebras (2008)
  • Author(s): 毛利出
  • Organizer: 空間の代数的・幾何的モデルとその周辺2008
  • Place of Presentation: 信州大学、長野
  • Year and Date: 2008-09-12
• [Presentation] The moduli spaces of non-thick irreducible representtations for the free group of rank 2 (2008)
  • Author(s): 中本和典
  • Organizer: 第41回環論および表現論シンポジウム
  • Place of Presentation: 静岡大学理学部
  • Year and Date: 2008-09-06
• [Presentation] The moduli spaces of non-thick irreducible representations for the free group of rank 2 (2008)
  • Author(s): 中本和典
  • Organizer: 第41回環論および表現論シンポジウム
  • Place of Presentation: 静岡大学理学部
  • Year and Date: 2008-09-06
• [Presentation] Symmetry and Asymmetry of the Euler Form (2008)
  • Author(s): Izuru Mori
  • Organizer: Comm. Algebra: Connections with Algebraic Topology and Representation Theory
  • Place of Presentation: Lincoln, NE, U.S.A.
  • Year and Date: 2008-05-20
• [Presentation] Symmetry and Asymmetry of the Euler Form (2008)
  • Author(s): Izuru Mori
  • Organizer: Commutative Algebra : Connections with Algebraic Topology and Representation Theory
  • Place of Presentation: Lincoln, NE, U.S.A.
  • Year and Date: 2008-05-20
• [Presentation] An Introduction to noncommutative Algebraic Geometry (2007)
  • Author(s): 毛利出
  • Organizer: 第5回日中韓環論国際シンポジューム
  • Place of Presentation: 東京・青少年記念オリンピックセンター
  • Year and Date: 2007-09-12
• [Presentation] An Introduction to Noncommutative Algebraic Geometry (2007)
  • Author(s): 毛利 出
  • Organizer: 第5回日中韓環論国際シンポジューム
  • Place of Presentation: 東京・青少年記念オリンピックセンター
  • Year and Date: 2007-09-12
• [Book] World Scientific Publishe (2009)
  • Author(s): Masahisa Sato
  • Total Pages: 289
  • Publisher: Ring Theory 2007 (Proceedings of the Fifth China-Japan-Korea Conference)
• [Book] Ring Theory 2007 Proceedings of the Fifth China-Japan-Korea Conference (2009)
  • Author(s): Masahisa Sato
  • Total Pages: 283
  • Publisher: World Scientific Publisher
• [Remarks] ホームページ等
  • URL: http://fuji.cec.yamanashi.ac.jp/~sato/
• [Remarks]
  • URL: http://fuji.cec.yamanashi.ac/~sato/
• [Remarks]
  • URL: http://fuji.cec.yamanashi.ac.jp/~sato/